Multi-linear Variable Separation Approach And Localized Excitations In Nonlinear Systems

Nowadays, with the gradual improvement of linear theory, nonlinear science has developed vigorously in the various research fields and become a focus of study. Therefore, it is unavoidable to meet a variety of nonlinear equations in the study process, and for solving the nonlinear equations undoubtedly become the key to the nonlinear scientific research, but also the difficulties of nonlinear study. Unlike linear equations, as the failure of the linear superposition principle, we didn't yet give the general solutions of nonlinear systems. A special solution can be given by one or several ways, but is not usually a way to various types of special solution. Thus, there is no uniform approach for solving nonlinear systems.Through the efforts of many scientists, people have already built up and developed a lot of effective methods for solving nonlinear systems. The multi-linear variable separation approach is a part of them, it has real meaning of variable separation. So far, the multi-linear variable separation approach has already successfully solved the large numbers of 2+1-dimensional nonlinear systems, some 1+1-dimensional and 3+1-dimensional nonlinear systems. And it has been already successfully applied to differential difference system as well. This paper mainly discussed variable separation techniques of the multi-linear variable separation approach, and thus gained a new multi-linear variable separation solution. By choosing a suitable arbitrary function, we got different new localized excitations of nonlinear systems. In addition, multi-linear variable separation approach can also be further promoted to the general multi-linear variable separation approach, and thus gained the general multi-linear variable separation solutions of nonlinear systems. These solutions contain more low-dimensional variable separation functions. By selecting appropriate arbitrary functions, we got different new localized excitations. Now the main contents of this paper are summarized as follows:Chapter one is an introduction, we mainly introduced the discovery and history of solitary waves and solitons, and summed up the current state of research. Then introduced mathematical research methods of nonlinear systems in brief, as well as gave out the research arrangement of this paper. In the second chapter, at first, we introduced development process and the current state on multi-linear variable separation approach. Second, we outline the general steps that multi-linear variable separation approach solves nonlinear systems. It is also quite important to apply the obtained variable separation solutions, which have several arbitrary functions(p and q). Then we applied multi-linear variable separation approach to 2+1-dimensional Boiti-Leon-Pemponelli (BLP) equation and 2+1-dimensional long wave dispersive equation, and discussed its new variable separation solutions and some new localized excitations. In general, if the choice of p and q for a variety of single-valued functions, we can obtain many types of single-valued localized excitations such as dromions, peakons, kink solitons, inverted solitons and so on.In the third chapter, the multi-linear variable separation approach can be promoted to the general multi-linear variable separation approach, getting the general multi-linear variable separation solutions of some nonlinear systems. These solutions contained more low-dimensional variable separation functions. By choosing the appropriate arbitrary functions, we got more abundant localized excitations.Finally, some main and important results as well as future research topics are given in the chapter.